Abstract

We present a detailed analysis of the modulational instability of the zone-boundary mode for one and higher-dimensional Fermi–Pasta–Ulam (FPU) lattices. Following this instability, a process of relaxation to equipartition takes place, which we have called the Anti-FPU problem because the energy is initially fed into the highest frequency part of the spectrum, at variance with the original FPU problem (low frequency excitations of the lattice). This process leads to the formation of chaotic breathers in both one and two dimensions. Finally, the system relaxes to energy equipartition on time scales which increase as the energy density is decreased. We show that breathers formed when cooling the lattice at the edges, starting from a random initial state, bear strong qualitative similarities with chaotic breathers.

Received 21 September 2004Accepted 11 December 2004Published online 28 March 2005

Lead Paragraph: Several nonlinear physical systems exhibit modulational instability, which is a self-induced modulation of the steady state resulting from a balance between nonlinear and dispersive effects. This phenomenon has been studied in a large variety of physical contexts: fluid dynamics, nonlinear optics and plasma physics. The Fermi–Pasta–Ulam (FPU) lattice is an extremely well-suited model system to study this process. Both the triggering of the instability and its further evolution can be studied in detail, exciting initially high-frequency modes. The original FPU problem was casted instead in the context of long wavelengths. This is why we call the process we analyze in this paper, the anti-FPU problem because of the analogy with the seminal FPU numerical simulation. At variance with the appearance of (m)KdV-solitons in the FPU original problem, in this process the pathway to equipartition leads to the creation of localized objects that are chaotic breathers. Similar localized structures emerge when cooling the lattice at the edges, starting from thermalized initial states.

Acknowledgments:

First, we would like to thank D. K. Campbell, P. Rosenau, and G. Zaslavsky for the opportunity they gave us to contribute to this Chaos issue celebrating the anniversary of the FPU experiment. Then, we express our gratitude to all our collaborators in this field: J. Barré, M. Clément, T. Cretegny, S. Lepri, R. Livi, P. Poggi, and A. Torcini. We also thank N. J. Zabusky for useful exchanges of information and Zhanyu Sun for an important comment. This work is part of the contract COFIN03 of the Italian MIUR Order and chaos in nonlinear extended systems. R.K. is supported by NATO reintegration Grant No. FEL.RIG.980767 and USA CRDF award No. GP2-2311-TB-02. S.R. thanks ENS Lyon for hospitality and financial support.